IR-TST 2012 Problem 16

Suppose $p$ is an odd prime number. We call the polynomial $f(x)=\sum_{j=0}^n a_jx^j$ with integer coefficients $i$ -remainder if $\sum_{p-1|j,j>0}a_{j}\equiv i\pmod{p}$ . Prove that the set $\{f(0),f(1),...,f(p-1)\}$ is a complete residue system modulo $p$ if and only if polynomials $f(x), (f(x))^2,...,(f(x))^{p-2}$ are $0$ -remainder and the polynomial $(f(x))^{p-1}$ is $1$ -remainder.

Proposed by Yahya Motevassel